Not knowing what you are sure of: e.g. “π + e is irrational.”

It is as unexceptionable as philosophical principles ever are that we can claim to have knowledge if, in addition to belief, we have the proper sort of justification. It seems we can have excellent justification for believing that that π + e  is irrational, leading to high confidence that it is. Yet no mathematician would say that we know the sum is irrational. Why is that?

(I ignore two other necessary conditions for knowledge: truth and not being “defeated.” Neither is relevant because I am here only interested in the making of a knowledge claim, not whether the claim is true.)

Neither π nor e is rational (representable as p/q,  p and q both integers, q not 0). In fact, neither is “algebraic” (roots of a non-0 polynomial equation with rational coefficients) The probability that a random real number is algebraic is 0, and even among the algebraics, rational numbers are, naively speaking, rare. For π + e  to be rational, something very weird has to be going on. The never-infinitely-repeating decimal representation of π would have to add together to the never-infinitely-repeating decimal representation of e to give a sum whose decimal expansion did repeat forever. 

Now, it is not hard to find irrational numbers whose sum is rational, π and –π, for example. Their decimal expansions cancel digit by digit to produce repeating 0s.  To produce a rational sum the digits of  π and e would have to dovetail in some more complex way, starting at some decimal place, and continuing out infinitely from there to yield periodic repetition. That sounds to me as if it would be a miracle.

So should we be confident that π + e  is irrational? We can surely be as confident as we are of death, taxes, that the sun will rise tomorrow, and that we could not, even if we tried hard, leap over tall buildings in a single bound. This confidence does not arise from any suspect source: religious dogma, partisan politics, personality idiosyncrasy, or from a pure act of will to believe because, after all, one must believe in something.  

We believe that this sum is irrational because of our mathematical knowledge. Indeed, I speculate that the more math someone knows the more confident she will be that π + e  is irrational. So why don’t we, or at least they who know a lot of the math in this vicinity, know that the sum is irrational?

I think the obvious answer is the right one, or at least the first big part of the right one. We demand more on the justification side for mathematical propositions because that more is so often available. In math we can prove things, and prove them in a way that leaves no, or almost no, room for rational dissent. We can give arguments the strength of which is unavailable in other branches of human inquiry. Such a good standard being available, it has been universally adopted. 

That is why mathematicians say, “We don’t know whether π + e  is irrational,” when they are very confident, on very good basis, that it is irrational. As good as that basis is, it is not good enough for the justification of a mathematical proposition because it isn’t proof. Occasionally a mathematician might push the point to rhetorical extravagance by saying things like, “we have no idea whether π + e is irrational.”  Of course, they have an idea, but they think it good pedagogy so to emphasize the importance of proof.

What I have said up to this point is, I think, mostly right, but also partly wrong in leaving something important out. Suppose that you arrive at work and a colleague rushes up to you to say, “the question of the rationality of π + e  has been settled.”  Let us, now, pause outside of time for you to consider whether you expect that the proof of which you are about to be informed to establish the rationality or the irrationality of the sum. Your first reaction, for the reasons canvassed above, would presumably be that it has been shown irrational. Yet, you consider the difficulty of proving things about a sum of irrationals. Mathematicians have long commented on their being nothing to “get hold of” for this particular problem. Yet, if your friend is right, there must have been something to be gotten hold of after all. The number world is not quite as everyone thought it was. In proportion as you thought that the proof might be impossible, the proof’s existence suggests that there is something bizarre going on, and that bizarreness might well be of just that sort that would permit these two decimal expansions when added together to start cycling. You might still be willing to give good odds that the revealed proof would be of irrationality not rationality, but those odds would, I think, be well shorter that what you would have offered before you heard of the existence of the proof. 

 

Proof is not only the standard for justification in math, it is the talisman that fends off the threat of weirdness  (when it doesn’t establish the weirdness).  

Mathematical truth, especially in the general vicinity of this proposition, has, often enough, been weird as measured against our prior intuitions. If we think it would amount to a miracle for π and e to add together to a rational number, we still are in want of assurance that no miracle is hidden just out of sight. We need to understand the structure in the way that proof supplies, and only proof supplies. (OK, sometimes proof doesn’t much show why, but merely that. Still, even that is something.) 

Consider this position:

If you think you have any reasonable basis for believing that ‘π + e  is irrational’ is probably true, you are mistaken. You cannot responsibly apply probabilities here, because you don’t know enough of the lay of the land. You seize on some facts within your cognizance, but all the important facts may well be unknown to you. You can have nothing more than a hunch supported by a woefully incomplete understanding. You may have a basis for your belief, but it is not a good basis. There is no basis short of proof that a rational person should credit in this case.

I think this is too strong. We do have a good basis for thinking the sum irrational. Still, there is something to this objection. It is not just that we say that a good basis that falls short of proof is not good enough because proof is available in math and so moves the bar up to that level. It is also the case that what would count as a good basis in other domains just is not as good in the world of numbers, parts of which are like Kansas, but other parts like Oz. It is not only that proof is a better tool; it is that the work to be done needs that better tool.